Can we square a distance?
I have 2 numbers, one of them is larger than the other.
If I place these numbers on a number line, then the distance between them is 3.
But if I multiply these numbers by itself to form 2 other numbers, and place these 2 other numbers on another number line, will the distance between them always be larger than 3?
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3 solutions
Great work! All we have to do is to find a counterexample to disprove this (universal) claim.
Can you find all possible solutions of these two numbers that satisfy all these properties, namely
∣ x − y ∣ = 3 and ∣ x 2 − y 2 ∣ > 3 ?
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There will be 4 pairs of ( x , y ) that are ( 2 , − 1 ) , ( − 1 , 2 ) , ( 1 , − 2 ) and ( − 2 , 1 )
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No, I'm not just looking for integer solutions only. Yes, you have written up all the pairs of integer solutions, but are there any other non-integer solutions as well?
Let the numbers be n and n + 3
all the numbers where n ∈ [ − 2 , − 1 ] will serve as counter examples
x 2 − y 2 = ( x − y ) ( x + y )
So as x − y = 3 , for ∣ x 2 − y 2 ∣ ≤ 3 we need ∣ x + y ∣ ≤ 1
So all ordered solutions ( x , y ) where x > y can be expressed as ( n , n − 3 ) for all n ∈ [ 1 , 2 ]
Let the 2 numbers be − 2 and 1 having a distance of 3 units.
Then, the new numbers formed will be 4 and 1 which have a distance of 3 units again.
Therefore, For two numbers having a distance of 3 units, it is not necessary that their squares will also have distance more than 3 units.